Zigzags in Turing Machines

Abstract

Abstract. We study one-head machines through symbolic and topological dynamics. In particular, a subshift is associated to the system, and we are interested in its complexity in terms of realtime recognition. We emphasize the class of one-head machines whose subshift can be recognized by a deterministic pushdown automaton. We prove that this class corresponds to particular restrictions on the head movement, and to equicontinuity in associated dynamical systems.Keywords: Turing machines, discrete dynamical systems, subshifts, formal languages.We study the dynamics of a system consisting in a finite automaton (the head) that can write and move over an infinite tape, like a Turing machine. We use the approach of symbolic and topological dynamics. Our interest is to understand its properties and limitations, and how dynamical properties are related to computational complexity. This approach was initiated by Kůrka in [1] with two different topologies: one focused on the machine head, and the other on the tape. The first approach was further developed in [2,3]. More recently, in [4,5], a third kind of dynamical system was associated to Turing machines, taking advantage of the following specificity: changes happen only in the head position whilst the rest of the configuration remains unaltered. The whole evolution can therefore be described by the sequence of states taken by the head and the symbols that it reads. This observation actually yields a factor map between Kůrka's first dynamical system and a one-sided subshift.In [4], it has been proved that machines with a sofic subshift correspond to machines whose head makes only bounded cycles. We prove here a similar characterization of machines with a shift that can be recognized by a deterministic pushdown automaton. Moreover, we establish links between these two properties and equicontinuity in all three spaces.In the first section, we recall the definitions and fundamental results. The second section is devoted to defining the different dynamical systems associated to one-head machines, and to stating basic results about equicontinuity within ⋆

The Group of Reversible Turing Machines

“…In [22], two dynamical systems were associated to a Turing machine, one with a 'moving tape' and one with a 'moving head'. After that, there has been a lot of study of dynamics of Turing machines, see for example [16,30,21,13,12,17,1]. Another connection between Turing machines and dynamics is that they can be used to define subshifts.…”

On relations between properties in transitive Turing machines

“…These two different topologies are not equivalent; therefore, a given machine may have different properties depending on the considered topological model, as Kůrka established. The seminal work of Kůrka inspired a large line of research that considers properties such as immortality [14,15,17], entropy [13,16,18,21], equicontinuity [9], periodicity [5,6,17,18], transitivity and minimality [6,12].…”

The Transitivity Problem of Turing Machines

“…The appropriate point of view is to study Turing machines as proper dynamicals systems as introduced by Moore [12] and further developed by Kůrka [9] who formalized two topologies associated to Turing machines and establishes several of its properties. Following that trend, several dynamical properties of Turing machines were recently studied: periodicity [2,3,8], entropy [6,13] and equicontinuity [4]. Here we focus on topological transitivity: the existence of a point whose orbit passes close to every other point of the space.…”